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tfp.substrates.jax.math.psd_kernels.FeatureScaled

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Kernel that first rescales all feature dimensions.

Inherits From: FeatureTransformed, AutoCompositeTensorPsdKernel, PositiveSemidefiniteKernel

Given a kernel k and scale_diag and inputs x and y, this kernel first rescales the input by computing x / scale_diag and y / scale_diag, and passing this to k.

With 1 feature dimension, this is also called Automatic Relevance Determination (ARD) [1].

References

[1]: Carl Edward Rasmussen and Christopher K. I. Williams. Gaussian Processes for Machine Learning. Section 5.1 2006. http://www.gaussianprocess.org/gpml/chapters/RW5.pdf

kernel PositiveSemidefiniteKernel instance. Inputs are rescaled and passed in to this kernel. Parameters to kernel must be broadcastable with scale_diag.
scale_diag Floating point Tensor that controls how sharp or wide the kernel shape is. scale_diag must have at least kernel.feature_ndims dimensions, and extra dimensions must be broadcastable with parameters of kernel. This is a "diagonal" in the sense that if all the feature dimensions were flattened, scale_diag acts as the inverse of a diagonal matrix.
inverse_scale_diag Non-negative floating point Tensor that is treated as 1 / scale_diag. Only one of scale_diag or inverse_scale_diag should be provided. Default value: None
validate_args If True, parameters are checked for validity despite possibly degrading runtime performance
name Python str name prefixed to Ops created by this class.

batch_shape Shape of a single sample from a single event index as a TensorShape.

May be partially defined or unknown.

The batch dimensions are indexes into independent, non-identical parameterizations of this PositiveSemidefiniteKernel.

dtype DType over which the kernel operates.
feature_ndims The number of feature dimensions.

Kernel functions generally act on pairs of inputs from some space like

R^(d1 x ... x dD)

or, in words: rank-D real-valued tensors of shape [d1, ..., dD]. Inputs can be vectors in some R^N, but are not restricted to be. Indeed, one might consider kernels over matrices, tensors, or even more general spaces, like strings or graphs.

inverse_scale_diag

kernel Base kernel to pass transformed inputs.
name Name prepended to all ops created by this class.
parameters Dictionary of parameters used to instantiate this PSDKernel.
scale_diag

trainable_variables

transformation_fn Function that preprocesses inputs before handing them to kernel.
validate_args Python bool indicating possibly expensive checks are enabled.
variables

Methods

apply

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Apply the kernel function pairs of inputs.

Args
x1 Tensor input to the kernel, of shape B1 + E1 + F, where B1 and E1 may be empty (ie, no batch/example dims, resp.) and F (the feature shape) must have rank equal to the kernel's feature_ndims property. Batch shape must broadcast with the batch shape of x2 and with the kernel's batch shape. Example shape must broadcast with example shape of x2. x1 and x2 must have the same number of example dims (ie, same rank).
x2 Tensor input to the kernel, of shape B2 + E2 + F, where B2 and E2 may be empty (ie, no batch/example dims, resp.) and F (the feature shape) must have rank equal to the kernel's feature_ndims property. Batch shape must broadcast with the batch shape of x2 and with the kernel's batch shape. Example shape must broadcast with example shape of x2. x1 and x2 must have the same number of example
example_ndims A python integer, the number of example dims in the inputs. In essence, this parameter controls how broadcasting of the kernel's batch shape with input batch shapes works. The kernel batch shape will be broadcast against everything to the left of the combined example and feature dimensions in the input shapes.
name name to give to the op

Returns
Tensor containing the results of applying the kernel function to inputs x1 and x2. If the kernel parameters' batch shape is Bk then the shape of the Tensor resulting from this method call is broadcast(Bk, B1, B2) + broadcast(E1, E2).

Given an index set S, a kernel function is mathematically defined as a real- or complex-valued function on S satisfying the positive semi-definiteness constraint:

sum_i sum_j (c[i]*) c[j] k(x[i], x[j]) >= 0

for any finite collections {x[1], ..., x[N]} in S and {c[1], ..., c[N]} in the reals (or the complex plane). '*' is the complex conjugate, in the complex case.

This method most closely resembles the function described in the mathematical definition of a kernel. Given a PositiveSemidefiniteKernel k with scalar parameters and inputs x and y in S, apply(x, y) yields a single scalar value.

Examples

import tensorflow_probability as tfp; tfp = tfp.substrates.jax

# Suppose `SomeKernel` acts on vectors (rank-1 tensors)
scalar_kernel = tfp.math.psd_kernels.SomeKernel(param=.5)
scalar_kernel.batch_shape
# ==> []

# `x` and `y` are batches of five 3-D vectors:
x = np.ones([5, 3], np.float32)
y = np.ones([5, 3], np.float32)
scalar_kernel.apply(x, y).shape
# ==> [5]

The above output is the result of vectorized computation of the five values

[k(x[0], y[0]), k(x[1], y[1]), ..., k(x[4], y[4])]

Now we can consider a kernel with batched parameters:

batch_kernel = tfp.math.psd_kernels.SomeKernel(param=[.2, .5])
batch_kernel.batch_shape
# ==> [2]
batch_kernel.apply(x, y).shape
# ==> Error! [2] and [5] can't broadcast.

The parameter batch shape of [2] and the input batch shape of [5] can't be broadcast together. We can fix this in either of two ways:

Fix #1

Give the parameter a shape of [2, 1] which will correctly broadcast with [5] to yield [2, 5]:

batch_kernel = tfp.math.psd_kernels.SomeKernel(
    param=[[.2], [.5]])
batch_kernel.batch_shape
# ==> [2, 1]
batch_kernel.apply(x, y).shape
# ==> [2, 5]
Fix #2

By specifying example_ndims, which tells the kernel to treat the 5 in the input shape as part of the "example shape", and "pushing" the kernel batch shape to the left:

batch_kernel = tfp.math.psd_kernels.SomeKernel(param=[.2, .5])
batch_kernel.batch_shape
# ==> [2]
batch_kernel.apply(x, y, example_ndims=1).shape
# ==> [2, 5]

batch_shape_tensor

View source

Shape of a single sample from a single event index as a 1-D Tensor.

The batch dimensions are indexes into independent, non-identical parameterizations of this PositiveSemidefiniteKernel.

Args
name name to give to the op

Returns
batch_shape Tensor.

copy

View source

Creates a copy of the kernel.

Args
**override_parameters_kwargs String/value dictionary of initialization arguments to override with new values.

Returns
copied_kernel A new instance of type(self) initialized from the union of self.parameters and override_parameters_kwargs, i.e., dict(self.parameters, **override_parameters_kwargs).

matrix

View source

Construct (batched) matrices from (batches of) collections of inputs.

Args
x1 Tensor input to the first positional parameter of the kernel, of shape B1 + [e1] + F, where B1 may be empty (ie, no batch dims, resp.), e1 is a single integer (ie, x1 has example ndims exactly 1), and F (the feature shape) must have rank equal to the kernel's feature_ndims property. Batch shape must broadcast with the batch shape of x2 and with the kernel's batch shape.
x2 Tensor input to the second positional parameter of the kernel, shape B2 + [e2] + F, where B2 may be empty (ie, no batch dims, resp.), e2 is a single integer (ie, x2 has example ndims exactly 1), and F (the feature shape) must have rank equal to the kernel's feature_ndims property. Batch shape must broadcast with the batch shape of x1 and with the kernel's batch shape.
name name to give to the op

Returns
Tensor containing the matrix (possibly batched) of kernel applications to pairs from inputs x1 and x2. If the kernel parameters' batch shape is Bk then the shape of the Tensor resulting from this method call is broadcast(Bk, B1, B2) + [e1, e2] (note this differs from apply: the example dimensions are concatenated, whereas in apply the example dims are broadcast together).

Given inputs x1 and x2 of shapes

[b1, ..., bB, e1, f1, ..., fF]

and

[c1, ..., cC, e2, f1, ..., fF]

This method computes the batch of e1 x e2 matrices resulting from applying the kernel function to all pairs of inputs from x1 and x2. The shape of the batch of matrices is the result of broadcasting the batch shapes of x1, x2, and the kernel parameters (see examples below). As such, it's required that these shapes all be broadcast compatible. However, the kernel parameter batch shapes need not broadcast against the 'example shapes' (e1 and e2 above).

When the two inputs are the (batches of) identical collections, the resulting matrix is the so-called Gram (or Gramian) matrix (